Collective dynamics of excitons and exciton-polaritons in nanoscale heterostructures

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Dmitri Visnevski

To cite this version:

Dmitri Visnevski. Collective dynamics of excitons and exciton-polaritons in nanoscale heterostruc-tures. Other [cond-mat.other]. Université Blaise Pascal - Clermont-Ferrand II, 2013. English. �NNT : 2013CLF22368�. �tel-00914332�

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No_{d'Ordre: D. U. 2366}

UNIVERSITE BLAISE PASCAL

-CLERMONT-FERRAND

U. F. R. Science et Technologies

ECOLE DOCTORALE DES SCIENCES

FONDAMANTALES

No₇₅₃

THESE DE DOCTORAT

Pour accéder au grade de

Docteur d'université en sciences

de l'université Blaise Pascal - Clermont-Ferrand

Specialité : Science des matériaux

Défendue par

Dmitri VISNEVSKI

Collective dynamics of excitons

and exciton-polaritons

in nanoscale heterostructures

Soutenue publiquement le 09 Juillet 2013 Commission d'examen:

Rapporteurs : Pr. Mikhail Glazov

Pr. Fabrice Laussy

Examinateurs : Pr. Dmitry Yakovlev

Pr. Ivan Shelykh

Dr. Dmitry Solnyshkov Directeur de thèse : Dr. Guillaume Malpuech

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Acknowledgments

Like the quasi-particles in semiconductor structures, living their short but undoubt-edly useful life, cannot be treated outside the media, so are the scientists and their works, which could not be considered stand-alone, out of the context of the scientic society. The time I have been writing this manuscript, I was wondering at the ele-gant consistency and regularity which underlay the evolution of the physics. When I tried to investigate the cause of any discovery or invention, I always went deeper and deeper into the history of physics, and if I did not stop myself at will, I certainly would have had to parse the manuscripts of the ancient Greeks.

Thus, the work that I have done is nothing but a natural continuation to all what has been done before, and a lot of other people have contributed to it not less than I did. I want to mention here some of them, who have made an invaluable contribution to the development of me as a person and as a scientist. First of all, of course, I want to thank my family and particularly my father, who, sometimes against my will and laziness, has been introducing me into the world of physics and mathematics in the times of my adolescence. Then, I would like to thank my rst true scientic advisor Prof. Ivan Vladimirovich Ignatiev. The rst tentative steps in the semiconductor physics I did under his experienced and wise leadership.

Of course, one could hardly overestimate the inuence of my three doctorate supervisors, who, contrary to the Russian proverb about seven nannies, each being a bright personality, harmoniously complement each other and provided me with a full-edged supervision. Thus, for example, innite optimism, vitality and incredible scientic intuition of Dr. Guillaume Malpuech have been continuously inspiring me and driving me forward, when sensible criticism and I would say even skepticism of Prof. Nikolay Gippius stopped me and cooled me down just in time, preventing of the making serious mistakes. Dr. Dmitry Solnyshkov helped me a lot with my pressing problems, I was admired by his ability to understand my needs at a glance (and not only because we speak the same language), accurately get the point of the problem and to provide the solution very quickly.

Also I wish to thank other PhD-students and one post-doctoral researcher I have worked with: Hugo Flayac, Goran Pavlovic, Anton Nalitov and Hugo Terças. It was very pleasant to collaborate with them, since we often have been sharing the same "joys and sorrows". I would like to mention a great team of the European International Training Network "Spin-optronics". We have been passing a very good time on numerous conferences and meetings which were held in dierent, sometimes quite exotic places.

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Introduction

Contents

1.1 Excitons and Exciton-polaritons . . . 3

1.1.1 The basics. Band structure. . . 3

1.1.2 Excitons . . . 4

1.2 Nanostructures . . . 6

1.2.1 Quantum Wells . . . 6

1.2.2 Quantum Dots . . . 8

1.2.3 Microcavities. Cavity polaritons . . . 10

1.3 Bose-Einstein condensates . . . 14

1.3.1 Basics . . . 14

1.3.2 Polariton scattering. Semiclassical Boltzmann equations . . . 16

1.3.3 Bogoliubov theory and Gross-Pitaevskii equations . . . 18

Undoubtedly, the successes in the investigation of the microworld in the begin-ning of XX century have provoked a rapid development of the quantum theory. Together with the works of Einstein on theory of relativity it has made a revolution in the conception of theoretical physics. New theories, whose progress was bounded just by the imagination of scientists, developed rapidly, often leaving the experi-ment behind. This way, for example, Bose-Einstein condensation was predicted by Einstein in 1925 [1]. Conditions for the formation of an atomic condensate were so extreme, that technologies allowed its observation only 70 years after its theoretical description[2].

One could say that the pioneer works on X-ray diraction in dierent crystal structures performed by von Laue, Bragg and others in early 1910s [3, 4, 5, 6] have initiated the modern solid-state physics (SSP). Since that time it has been continuously evolving during the XXth century. Progress in growth technologies has been allowing to create more and more complicated and perfect structures - starting with simple transistors in 1940s up to high-quality nanostructures nowadays. Most of them, based on the specics of heterojunctions between dierent materials, are eciently driving forward micro- and nanoelectronics, optoelectronics, spintronics etc.

In addition, it's important to say about one more aspect - information tech-nologies (IT). A lot of theoretical problems have no analytical solution. Previously,

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scientists had to make numerous approximations to solve them and this limited the applicability of solutions. The strong boom in the IT of the last 30 year has allowed to solve very wide range of problems with high accuracy for a reasonable time. In the solid-state physics numerical calculations are extremely important, so the progress in computer technologies strongly assists the development of the solid-state theory. Summarizing, it seems, that we live in the age where the development of science becomes less and less limited by objective constraints.

Huge number of particles, which compose solids, gives rise to numerous collec-tive excitation eects. They can be described in terms of quasiparticles (excitons, phonons, etc), which behave like real particles having its own eective mass, charge, spin etc, but which appear in media only. The solid medium strongly denes the behavior of quasiparticles, so tuning the medium properties one could strongly vary the properties of the quasiparticles under study. This is why nowadays there are so many dierent nanostructures. Solid-state physics provides a handy platform to study and develop dierent eects which could be hardly studied in other sys-tems. So, for example, considering very light quasiparticles (whose mass is 108_times

smaller than atomic mass) one could raise the critical temperature of Bose-Einstein condensation from microKelvins up to room temperature. Amazing properties of semiconductor nanostructures have inspired me to dedicate this work to them. Of course, modern solid-state physics is a huge scientic eld, but I want to believe that my studies will nd their small but useful place inside.

This manuscript is divided into 5 chapters. The rst chapter is an introduc-tion, where I will describe the basics aspects of nanostructure physics which are relevant to my work, such as basic facts about structures of dierent types, idea of Bose-Einstein condensation and basic mathematical tools for its description. Each of the four others chapters will be dedicated to a separate topic. In the second chapter I will discuss the interaction between exciton condensates and strong co-herent acoustic elds. I will show that at some conditions strong-coupling regime could be obtained what gives rise to a new-type quasiparticles. Chapter 3 will be dedicated to the lasing of quantum dots ensemble embedded into a cavity. The eect of the lasing amplication by ultrafast acoustic pulse will be described. In chapter 4, the multistability eect in the system of cavity polaritons will be dis-cussed. It will be shown that under strong resonant pumping of the ground state, the high-populated excitonic reservoir could appear, which strongly modies the ef-fective polariton-polariton interactions. And nally, in chapter 5 I will speak about polarization patterns in the condensates of indirect excitons. Also, I have added a small Appendix, where I discuss some features of numerical methods I used in my calculations.

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1.1. Excitons and Exciton-polaritons 3

1.1 Excitons and Exciton-polaritons

1.1.1 The basics. Band structure.

The propagation of the free carriers (electrons and holes) in the periodic potential of semiconductor lattice can be described by the Bloch-type wave-functions unk(r)

multiplied by plane wave [7]:

ψnk = eikrunk(r), (1.1)

where the function unk(r + R) = unk(r) has the same periodicity as crystal

structure. r is a radius-vector, k is called electron (hole) wave-vector and n is a band number. Thus, the carriers in semiconductor could be considered as quasiparticles with their own dispersion dened by the medium.

Figure 1.1: Band structure of GaAs for electrons with small wave-vector. Valence band consists of three subbands (SO, LH and HH). Eg is the width of the energy

gap, ∆SO is a split-o value

Hereinafter we will speak generally about semiconductors of zinc-blende type and particularly about GaAs. In such crystals the highest valence band is formed by p-shell electrons while the conduction band is formed by s-shell electrons. Electrons in p-states have angular momentum L equal to 1 so its projection Lz on the chosen

axis (let's call it Z axis) could be either 0 or ±1. Electrons with Lz = 0 are split

in energy from the bottom of the valence band and they form the so-called split-o band (SO). Total angular momentum J of a carrier is dened by the sum of its angular momentum L and its spin S. So, the split-o band is double degenerate when the spin is taken into account. The projection of J on chosen axis for the states with Lz = ±1 could take for values (Jz = ±1₂,±3₂). This gives rise to two

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double degenerate bands called light-hole band (LH) for Jz = ±1₂ and heavy-hole

(HH) band for Jz =±3₂ which show the same energy when carrier has zero

wave-vector k = 0 [8]. In our work we will not consider the SO band and will discuss LH and HH bands only.

The total angular momentum of s-type electrons which form the conduction band is dened by the electron spin only. Thus, there is only one double degenerate con-duction band. Band structure of GaAs for the carriers with small k is schematically illustrated on gure 1.1.

If a photon is propagating in a semiconductor and its energy is larger than the gap energy Eg, it could be absorbed by the medium and a free electron and

a hole would appear. Physical laws require the conservation of the wave-vector (kph= ke+ kh) and of the angular momentum. Taking into account the fact that photons have angular momentum equal to 1, the latter requirement provides certain selection rules:

J_ze+ J_zh =±1, 0. (1.2)

Selection rules for photon absorption are schematically shown in gure 1.2[8]. Circularly polarized photons with Jph

z = ±1 are shown by circle arrows, while

linearly polarized photons with Jph

z = 0 are plotted by straight double arrows.

Figure 1.2: Selection rules for light absorption. Black numbers show the z-projections of total angular momentum of carriers. Green arrows show the polariza-tion of the absorbed light when green numbers state relative intensity of transipolariza-tions. Also, an electron and a hole could recombine emitting a photon. The selection rules for emission are similar to ones for absorption. Thus by illuminating samples by polarized light one can create certain spin states in the system and by analyzing the polarization of emitted light, one can collect the information about spin states. This is what lies in the principles of the light-based control of spin.

1.1.2 Excitons

An electron interacts with a hole by the Coulomb force. If their relative distance

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1.1. Excitons and Exciton-polaritons 5 Schrodinger equation for the system could be written in a form:

( ˆ p2_e 2me + pˆ 2 h 2mh − e2 εre−h ) Ψ = EΨ. (1.3)

Here ˆpe,hare momentum operators for electron and hole, me,hare their eective

masses, Ψ is a wave-function of the system and E is the energy. The semiconductor medium is introduced in this equation by the dielectric constant ε. In such descrip-tion, the electron-hole pair forms a hydrogen atom-like system and one can speak about new quasi-particle - exciton [9]. The exciton energy spectrum is described by:

E_nX(kX) =− RX n2 + ~2_k2 X 2M , (1.4) RX = µe 4

2~2_ε2 is called exciton Rydberg, kX is an exciton wave-vector, M =

me + mh - its eective mass and µ =

( 1 me + 1 mh )₋₁

is the reduced mass. Here, the lowest energy of free electron and hole is taken as zero reference. Equation 1.4

describes the parabolic dispersion of excitons. Bohr radius of an exciton could be written as:

aX = ~

2_ε

µe2. (1.5)

Dielectric constant ε of GaAs is 13 which makes Bohr radius of an GaAs exciton much larger than the size of its unit cell (aX = 112 Å). Thus our assumption

about relative position of the electron and the hole is correct, and eective mass approximation (1.3) is legal. Such excitons with large Bohr radius are commonly found in crystal semiconductors and are called Wannier-Mott excitons as opposed to Frenkel excitons of small Bohr radius which could be found in molecular crystals. In this work we shall be dealing only with Wannier-Mott excitons.

The main property of excitons is their ability to interact with light (theory was developed rst in [10,11,12]) An exciton could be created by light as well as it could decay, emitting photon. Exciton-photon interaction is strongest at resonance. If we introduce into the Hamiltonian the exciton-photon interaction term, the eigenstates of the system at the wavevector where the dispersions cross each other would not be pure exciton and photon states, but a mixing between them. In fact we could speak about a new quasi-particle - exciton-polariton [7]. Thanks to light-matter interaction, an anticrossing between the two dispersion curves appears. However, in most bulk materials, exciton-photon interaction is weak, so the magnitude of the anticrossing is smaller than broadening of exciton and photon lines. In these cases, the eigenstates of system are almost completely excitonic or photonic and we can still speak about two independent quasi-particles in medium.

Sum of the total angular momentum of hole and electron forming the exciton is called for simplicity the exciton spin (SX_{). Its projection on the chosen axis can} take ve values: SX

z = ±2, ±1, 0. Selection rules and conservation laws here are

the same as for free electrons and holes, this is why excitons with SX

z =±2 do not

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1.2 Nanostructures

Growth technologies have been advancing very much during the last decades. This gave opportunity to fabricate dierent semiconductor heterostructures with very high quality interfaces. As a result, numerous nanoscale heterostructures appeared. The main idea is to strongly modify motion of carriers by heterotransitions. For example, placing a thin layer of one semiconductor with small band gap (for example GaAs) between two layers of another crystal with large band gap (AlAs) one can create a potential well for carriers in the growth direction (Z-axis). If the size of potential well is small enough, the motion of carriers along Z becomes quantized. Such structures are called Quantum wells or QWs. Particularly, if electrons and holes are trapped in the same layer, such QWs (or, rather, the heterojunctions) are said to be of the 1st type.

Electrons and holes in QWs form a quasi 2D system. Their motion in the plane is free, while they are quantized in the growth direction. Nowadays, there are plenty of dierent types of heterostructures possessing dierent eective dimensionality. However, in this work we will discuss only few of them.

1.2.1 Quantum Wells

As it was said, quantum wells are thin layers of one semiconductor sandwiched between thicker layers of another. The motion of the carriers becomes quantized in the Z direction and a number of energy subbands appears. Because each state trapped in QW has a xed nonzero kz, the degeneracy between LH and HH bands in Γ point is lifted and even with zero in-plane wave vector k_|| there is a gap in energy between the two bands. For an innite potential well one should expect the following expression for the width of this gap [13]:

∆E_lhΓ_−hh= (πn~) 2 2L2 QW ( 1 mlh − 1 mhh ) , (1.6)

where LQW is the width of the well and n is a number of state quantized in Z.

However, far from the Γ point, the eigenstates in the valence band are no longer pure heavy-hole or pure light-hole states, but a mixture of them. This gives rise to additional non-parabolicity of the valence band.

In quantum wells, electrons and holes could couple to excitons as well, but quan-tum connement of their motion imposes certain changes on their energy structure. Thus, in excitons formed by an electron and by a heavy hole, optically active states (SX

z =±1) are split from dark states (SXz =±2) by short-range electron-hole

ex-change interaction.

For spin dynamics of carriers in QWs one should consider two dierent cases: when carriers are free and when they are bound into excitons. The general mecha-nism of spin relaxation for free carriers is a Dyakonov-Perel mechamecha-nism of spin-orbit interaction. This gives rise to non-zero splitting between spin-states proportional to k. In bulk there is only one type of spin-orbit interaction - the Dresselhaus or bulk

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1.2. Nanostructures 7 inversion asymmetry (BIA)[14]. For electrons in a basis ( 1

2 − 1 2

)T

one can read a Hamiltonian term:

H_eD = β(σxkx(k2y− k2z) + σyky(kz2− k2x) + σzkz(k2x− ky2)), (1.7)

where β is a Dresselhaus coecient.

To obtain the corresponding term in Hamiltonian for electrons in QWs one should take an averaging over kz. For [001] direction of growth axis one could get:

H_2DeD = βe(σ+ke₋+ σ−k+e). (1.8)

Here (σ±, k±) = (σx, kx)± i(σy, ky).

Also in conned structures there is an additional energy splitting due to the structural inversion asymmetry (SIA) in the presence of electric elds also known as Rashba spin-orbit interaction[15]:

H_eR= γ(σ+(kye+ ikxe) + σ−(kye− ikex)). (1.9)

Here γ is Rasba coecient.

For holes the situation is a bit more complicated. In bulk, the hole spin relaxes very fast due to the mixing between heavy-hole and light-hole subbands [8,16]. In 2D systems, because of the degeneracy lifting near the Γ-point this mechanism is suppressed and holes are also aected by SOI. Several recent papers claim the cubic by k dependence of splitting[17,18]. However, to my knowledge it is the result of the group delusion and the Dresselhaus term in the Hamiltonian for 2D holes is linear by wave-vector k as it was stated in the paper of Rashba and Sherman [19]. The energy splitting between two subbands in the heavy-hole band reads as:

∆Ehh(k) = β ′ h ( π LQW )2 k. (1.10)

Because of the HH-LH subband splittings in 2D-systems, the lowest in energy exciton is formed from heavy-holes. The short-range exchange interaction between holes and electrons bound into an exciton causes a splitting in energy between bright and dark exciton states in quantum well for a value [13] (here and after we will consider the innite barrier model):

∆0 ≈ 9 16∆ 3D 0 ( EB E_B3D )2 _a3D B LQW . (1.11) Here ∆3D

0 is the corresponding splitting in bulk, EB3D,EB are the exciton binding

energies in bulk and in QW, a3D

B is a Bohr radius.

Also, thanks to the long-range electron-hole exchange interaction there is an additional splitting in energy between the longitudinal and transverse exciton states, which in 2D systems is linear with respect to the wave-vector of the exciton center of mass kX _[₂₀_]:

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∆LT = 3 16∆ 3D LT ( EB E3D B )2 |⟨χe,l|χh,l⟩|2a3DB kX, (1.12)

where χe,h,l are the single particle envelope functions describing the electron

and heavy-hole motion along the growth axis, and ∆3D

LT is a longitudinal-transverse

splitting for the bulk, which does not depend on kX_.

1.2.2 Quantum Dots

Another very important type of semiconductor nanostructures, in which motion of carriers is conned in all directions, are called quantum dots (QD). The his-tory of QDs starts in 1981 with the work of Ekimov and Onushchenko [21], who observed discrete lines in the absorption spectra of CuCl nanocrystals in a silicate glass matrix. However, the energy structure of such objects was very sensitive to the interfaces. Since that time, numerous methods of quantum dots growth appeared, but most popular nowadays is Stranski-Krastanov method of self-assembled quan-tum dots formation under molecular-beam epitaxy (MBE). This method is based on the formation of semiconductor droplets under the forces of surface tension. Then these droplets are covered by another wide-gap semiconductor. However, between the QDs there still remains a thin layer of the same material which is called wetting layer. Such growth method leads to the formation of a large ensemble of quantum dots. However, studies of single quantum dots selected from such ensemble, are also popular.

A scheme and a HRTEM image of quantum dots are presented on g. 1.3.

Figure 1.3: a) Schematic representation of selfassembled quantum dots. 1, 4 -buer layers of GaAs, 2 - InAs quantum dots, 3 - wetting layer, 5 - n-doped GaAs substrate. b) HRTEM image of single InAs quantum dot. Pictures are taken from [22,23]

As it was said, in QDs the motion of carriers is conned in all directions which results in the formation of a set of discrete states. Considering an approximation of the innite potential well for carriers and considering a parallelepiped QD with dimensions Lx,y,z, one can easily nd the values for the energy of the quantized

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1.2. Nanostructures 9 E_nQD = ~ 2_π2 2me,h ( n2_x L2 x + n 2 y L2 y + n 2 z L2 z ) , (1.13)

where nx,y,z are the quantum numbers of the eigenstates. Thus, the ground

exciton state has the energy:

EQD_X1 = ~ 2_π2 2µ ( 1 L2 x + 1 L2 y + 1 L2 z ) . (1.14)

Obviously it depends strongly on the size of the dot. The Stranski-Krastanov method assumes self-assembling of the quantum dots which leads to a statistical distribution of their sizes. As a result, there is a broadening of the QD spectra up to tens meV which is called inhomogeneous broadening (g. 1.4). Annealing of the structure at high temperature provokes the diusion of atoms and blurs the interfaces of the quantum dots. This procedure leads to a smoothing of the dots sizes and to a narrowing of their spectra. At the same time, the average ground state energy is increased.

Figure 1.4: Spectra of InAs/GaAs QDs photoluminescence for dierent values of the annealing temperature.

The main method of study of QDs is based on the analysis of their photolumi-nescence (PL). Investigation of the eects based on polarization properties of PL takes a large part in the QD eld, however, it is not the principal subject of this thesis, so we will not discuss the features determined by the spin dynamics of the carriers.

Nowadays, the quantum dots are considered as very promising objects. They are already widely exploited in light-emitting devices: light-emitting diodes and lasers. Recently, displays on quantum dots appeared. Also, they have been used as

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active media for solar cells, single electron transistors and even qubits for quantum computation.

1.2.3 Microcavities. Cavity polaritons

Connement of light in resonators has been studied since a very long time, and maybe the most well-known resonator is the Fabry-Perot resonator. The most used method is the connement in optical microcavities (MC). Nowadays, the microcavi-ties are usually composed of two Bragg mirrors situated parallel each to other with some spacing between. Each Bragg mirror presents a periodical structure of two semiconductors with dierent dielectric constants (g. 1.5)

Figure 1.5: Scheme of a microcavity. A number of quantum wells or quantum dots are embedded at the antinodes of the light mode. Image is taken from [24]

Tuning the thickness of layers, one can obtain the situation when the incident light is almost totally reected due to the interference inside the mirror. The spectral width of the total reection band is called stop-band. However, for two parallel mirrors, a dip may appear in the reection spectrum. This dip corresponds to the resonant frequency of the light mode conned inside the resonator. Due to the fact that cavity photons have xed wave-vector along the growth axis, their dispersion is parabolic for small in-plane wave-vectors. The main feature of microcavities is that photons could live inside very long time (up to tens picoseconds) thanks to numerous reections from the two Bragg mirrors. This requires the reection coecient to be very high, and this is why the Bragg mirrors are preferred over metallic ones, for example.

In general, the photonic dispersion law is linear versus its wave-vector and could be written in a form:

Eϕ=~kcϕ, (1.15)

where cϕis a speed of light in the medium. However, in the microcavities, where

the motion of photons is conned in the Z direction, the value of kz is xed and

one should express the total photon wave-vector as k = √k2

z+ k_||2, where k|| is an

in-plane wave-vector. Thanks to this, the dispersion in a case of small k_|| gets the standard parabolic form:

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1.2. Nanostructures 11

Figure 1.6: Reection spectra of the quantum wells in a microcavity measured for dierent values of detunings. The presence of a double dip is a signature of the transition to strong coupling regime. The image is taken from [25]

Eϕ∼

~2_k2

||

2mϕ

. (1.16)

The eective mass of cavity photon is given by mϕ= kz/2cϕand it usually takes

values of the order of 10−5_m_e_.

Usually, the microcavities are used to study the light-matter interaction and the number of quantum wells or quantum dots are embedded inside (g. 1.5). They are placed at the antinodes of the conned light mode to increase the strength of the interaction. If the light-exciton interaction is strong enough, one can obtain the strong coupling regime which is exhibited by anticrossing between the light and the exciton mode. In microcavities the strong-coupling regime for the rst time was observed by Claude Weisbuch et al.[25]. They studied the dip in the reectivity spectrum of the QWs embedded in MC and they found that at some conditions the dip was split (g. 1.6), which is a signature of the anticrossing of quantum levels because of their interaction.

To describe light-matter interaction in microcavities mathematically we could write the following Hamiltonian:

H =∑ k Eϕ(k) ϕ†_kϕk+ ∑ k Eχ(k) χ†_kχk+~ΩR ∑ k ( ϕ†_kχk+ χ†_kϕk ) . (1.17)

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Here ϕ†kand ϕk(χ†kand χk) are the creation and destruction operators for a photon (exciton). Eϕ(k) =~2k2/2mϕ+ δ and Eχ(k) = ~2k2/2mχ describe the parabolic

dispersions of photons and excitons. The ground state energy of the photon could be shifted with respect to the ground state energy of the exciton by a value of δ which is called detuning. The strength of the exciton-photon interaction is described by the so-called Rabi frequency ΩR.

In the two-component basis (pure exciton and pure photon states) the 1.17

Hamiltonian could be presented in a matrix form:

M = ( Eϕ(k) ~ΩR ~ΩR Eχ(k) ) . (1.18)

It can be easily diagonalized. As a result, one can write the in-plane dispersion expressions for the two new eigenstates:

EL(k) = 1 2 ( Eϕ(k) + Eχ(k)− √ [Eϕ(k)− Eχ(k)]2+ 4~2Ω2R ) , (1.19) EU(k) = 1 2 ( Eϕ(k) + Eχ(k) + √ [Eϕ(k)− Eχ(k)]2+ 4~2Ω2_R ) . (1.20)

Figure 1.7: Polariton dispersion calculated for a) negative, b)zero and c) positive detuning. Image is taken from [24]

As stated in1.1.2, the new eigenstates can be treated as new quasiparticles which are called in general exciton-polaritons or particularly cavity polaritons (because they appear in a microcavity). Polariton dispersion for dierent values of detuning is presented on g. 1.7.

The expression for the energy splitting between the lower and the upper polariton branches for zero in-plane wave vector reads:

∆EU L(0) =

√

δ2₊_~2_Ω2

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1.2. Nanostructures 13 If the broadening of polariton lines is smaller than this splitting, then the double dip in the reectivity spectrum could be observed. This is a practical criterion for the strong-coupling regime.

The polariton dispersion at small in-plane wave-vectors is almost parabolic and could be described by an eective mass whose value lies between mϕ and mχ. The

value of the mass strongly depends on the detuning. Thus, δ is a very important parameter in polaritonics. In practice, microcavities are grown that way that Bragg mirrors are not absolutely parallel, but slightly mutually inclined (because there is a wedge in the cavity thickness due to a special growth procedure). This allows to tune the energy of photonic ground state and the δ by studying dierent points on the cavity surface.

As it was said before, the long-range exchange interaction between electron and hole coupled to an exciton gives rise to a so-called longitudinal-transverse (or TE-TM) energy splitting between dierently polarized exciton states. However, for cavity polaritons there is an additional TE-TM splitting originating from the one of cavity photons. Since now we should no longer consider the model of two coupled oscillators (photons and excitons) but a model of four oscillators (TE- and TM-polarized excitons and photons), each of them has its own eective mass and a bare dispersion: E_ϕT E(k) = ~ 2_k2 2mT E_ϕ + δ, E T E χ (k) = ~2_k2 2mT E χ , (1.22) E_ϕT M(k) = ~ 2_k2 2mT M ϕ + δ, E_χT M(k) = ~ 2_k2 2mT M χ . (1.23)

This leads to 4 polariton branches:

ET E_L = E T E ϕ + EχT E 2 − 1 2 √( E_ϕT E − ET E χ )2 + 4~2_Ω2 R, (1.24) E_LT M = E T M ϕ + EχT M 2 − 1 2 √( ET M ϕ − EχT M )2 + 4~2_Ω2 R, (1.25) ET E_U = E T E ϕ + EχT E 2 + 1 2 √( E_ϕT E − ET E χ )2 + 4~2_Ω2 R, (1.26) E_UT M = E T M ϕ + EχT M 2 + 1 2 √( E_ϕT M − ET M χ )2 + 4~2_Ω2 R. (1.27)

For the lower polariton branch in a case of small wave-vectors, the additional energy coming from the TE-TM splitting could be written in a basis of (1, −1)T_:

∆ELT = ( 0 βLT(ky− ikx)2 βLT(ky+ ikx)2 0 ) , (1.28)

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Physically, it means that at some certain k one can consider an eective in-plane magnetic eld which causes "Zeeman" splitting between dierently polarized polaritons (we describe the splitting between polarizations as being caused by an eective eld). The direction of this eld changes as we change the direction of the wave-vector.

1.3 Bose-Einstein condensates

1.3.1 Basics

The story has started in 1925 by the work of Einstein [1]. Basing on the theoretical work of Bose [26], he predicted a new phase transition in a system of noniteracting bosons. Let us consider a ν-dimensional bosonic gas. Its distribution function is:

fB(k, T, µ) = 1 exp ( E(k)−µ kBT ) − 1. (1.29)

Here E (k) is a dispersion of a boson, µ is a chemical potential, T is a temperature and kB - the Boltzmann constant. It's necessary to notice that 1.29 requires µ to

be negative if E(0) = 0.

To obtain the total density of particles one should integrate1.29over all states in the reciprocal space:

n (T, µ) = 1

(2π)ν ∫ +∞

0

fB(k, T, µ) dνk. (1.30)

Also we can extract the density of particles in the ground state from the integral:

n0(T, µ) = lim R→+∞ 1 Rν 1 exp ( −µ kBT ) − 1. (1.31)

Where the size of the system R tends to innity. So we have: n (T, µ) = n0+ 1 (2π)ν ∫ fB(k, T, µ) dνk. (1.32)

So far as µ is negative and it grows with the number of particles in system, there could be a nite nc(T, 0)for which the chemical potential turns to zero and it seems

to be the maximum particle density in the system:

nc(T, 0) = lim µ→0 1 (2π)ν ∫ fB(k, T ) dνk. (1.33)

For the 3D-case this integral converges and could be calculated analytically, while for the cases of ν < 3 it diverges. For a parabolic dispersion of bosons, the calculated integral gives:

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1.3. Bose-Einstein condensates 15 nc(T ) = ζ ( 3/₂) (mkBT 2π~2 )3/₂ , (1.34)

where ζ(3/₂)_{is a Riemann Zeta function. Also, for a xed density of particles} there exists a nite temperature Tc:

Tc= ( n ζ(3/2) )2/3 h2 2πmkB . (1.35)

But what happens in the system when we exceed the nc or when we cool our

system below the Tc? It was proposed that extra particles would go to the lowest

level which was not considered in 1.33. Indeed, when the chemical potential goes to zero, there is a divergence in eq. 1.31, so we can say that density of particles condensed in the ground state is:

n0(T ) = n(T )− nc(T ). (1.36)

This process was called Bose-Einstein condensation of particles. So far as it is easier to manipulate the temperature of the gas than the number of particles, we will discuss generally Tc. For example, for the gas of rubidium atoms the critical

temperature has the order of hundreds of nanoKelvins. The Bose-Einstein conden-sation at such extremely low temperatures was observed for the rst time in 1995 by Eric Cornell and Carl Wieman at the University of Colorado at Boulder [2]. For this work, Cornell, Wieman, and Wolfgang Ketterle received the Nobel Prize in Physics in 2001.

For the systems with reduced dimensionality (2D, 1D, 0D) the integral 1.33

diverges and the logic described above is not valid anymore. However, real semi-conductor nanostructures which provide 2(1,0)D systems have nite sizes. It means that in this case integral 1.33 converges as well, even for smaller dimensionalities, and the quasi-condensation is possible.

Also, the possibility of the low-temperature phase transition to the superuid state of the 2D bosonic system was considered by Berezinskii, Kosterlitz and Thou-less [27,28] in the case of interacting bosons. The BKT transition was observed in a 2D gas of Rubidium atoms in 2006 [29].

As it could be seen in1.35, Tcis inversely proportional to the mass of particles.

The semiconductor systems provide numerous bosonic quasiparticles which are much lighter than atoms, what could simplify their condensation conditions. For example, the mass of an exciton is approximatively 105 _{times less than the mass of rubidium}

atom. Moreover, cavity polaritons could be 109 _{times lighter than atoms, which}

in theory leads to the room temperature values of Tc. Condensation of polaritons

was rst demonstrated very recently - in 2006 in the work of Kasprzak et. al. [30]. Fig. 1.8 shows the angle distribution of the cavity emission and its dispersion for dierent values of non-resonant pumping density. When the number of polaritons

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exceeds the value of nc, the phase transition occurs which provides sharp peaks both

in angular distribution and in dispersion.

Also in 2012, the spontaneous coherence in a cold exciton gas was reported [31] what could be the evidence of the exciton condensation.

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Figure 1.8: The angle distribution of the emission intensity (upper panels) and the dispersion of emission (lower panels) for dierent values of the density of excitation. Image is taken from [30]

1.3.2 Polariton scattering. Semiclassical Boltzmann equations

In classical statistics, the dynamics of the distribution function (the k-state popu-lation nk) is typically described by the Boltzmann equations:

dnk dt = ∑ k′ Wk′→knk′− nk ∑ k′ Wk→k′, (1.37)

where the sums go over all other states dierent from k and Wk′→k′′ is a total

scattering rate between k′ _{and k}′′ _states.

However, in our consideration the quantum properties of particles are very important. Uhlenbeck and Gropper in 1932 proposed a new concept of kinetic equations[32] which takes into account the quantum nature of particles. They are called semi-classical Boltzmann equations and have a form for bosons:

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1.3. Bose-Einstein condensates 17 dnk dt = (1 + nk) ∑ k′ Wk′→knk′− nk ∑ k′ Wk→k′(1 + nk), (1.38)

and for fermions:

dnk dt = (1− nk) ∑ k′ Wk′→knk′− nk ∑ k′ Wk→k′(1− nk). (1.39)

Unlike the atoms, which could live almost innitely long (there is a certain escape rate for atomic condensates as well, which is even used for their evaporative cooling) and therefore could be easily thermalized at any temperature, excitons and exciton-polaitons have nite lifetime. This is why, to keep the number of particles in the system constant one should have a permanent source of particles. In semiconductor nanostructures it could be obtained by a non-resonant (coherent or not) pumping of the high-energy states. Then, the quasiparticles could thermalize by scattering on phonons, if their lifetime is much longer than the eective times of scattering processes. We can introduce phenomenologically the pumping (Pk) and the lifetime

(Γk) to 1.38: dnk dt = Pk− nkΓk+ (1 + nk) ∑ k′ Wk′→knk′− nk ∑ k′ Wk→k′(1 + nk). (1.40)

There are few main mechanisms of polariton scattering: scattering on the struc-tural disorder, polariton-phonon scattering and polariton-polariton scattering. In the processes of the rst type, the propagating polaritons interact with the struc-tural disorder of the system. However, this interaction can not change the absolute value of the polariton wave-vector and the scattering occurs on an elastic circle in reciprocal space. One could take this scattering into account by considering a cylindrically symmetrical distribution of polaritons, which is the main result of such scattering.

The second scattering mechanism - polariton-phonon scattering - consists of two parts: interaction with 2D optical and 3D acoustic phonons. Optical phonons carry relatively large energies, and they are responsible for the initial polariton relaxation. However, when polaritons relax to the energies below 20 meV (the activation energy of an LO phonon), this scattering process becomes inecient. After that, polaritons could relax further by the interaction with acoustic phonons. This interaction is mediated by the deformation potential and involves relatively small amounts of energy.

Finally, polariton-polariton scattering can be very a ecient process of polariton thermalization, however, this process conserves the total energy of the polariton system. Scattering rate here depends strongly on polariton density. This process is responsible for an important non-linear optical eect called Optical Parametric Oscillator (OPO) observed rst for polaritons by P. Savvidis et. al. in 2000 [33] and described theoretically by Ciuti et. al. in [34].

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Figure 1.9: Experimental observation of the bottleneck eect. The highest intensity of emission is observed from the energies which correspond to the bottleneck region. Image is taken from [37]

As it was said, in order to obtain the condensation of polaritons in the ground state, polaritons should relax faster than they decay. However, due to the fact that most scattering goes via the excitonic part of the polariton and that polaritonic dispersion becomes very steep at small k-vectors, there is an eective deceleration of relaxation processes at the region called "bottleneck" [35,36,37]. Experimentally it can be observed by an enhanced intensity of cavity photoluminescence at certain

k(g. 1.9). In order to overcome the bottleneck eect, high densities of pumping or

cavities with longer lifetimes and positive detuning between photonic and excitonic fractions are used.

The general behavior of polaritonic system is schematically illustrated on g.1.10.

1.3.3 Bogoliubov theory and Gross-Pitaevskii equations

The ideal BEC consists of non-interacting bosons. However, real bosons are interacting. The Hamiltonian for the system of uniformely distributed weakly-interacting bosons could be written in second quantization terms as:

ˆ H =∑ k ~2_k2 2m â † kâk+ α 2L3 ∑ k1,k2,q ˆ a†_k 1+qâ † k2−qâk1aˆk2, (1.41)

where ˆak, ˆa†k are annihilation and creation operators for a particle with k

wave-vector, α is an interaction constant and L is a size of a system.

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1.3. Bose-Einstein condensates 19 103 104 105 106 1.455 1.460

Ω

R

X

bottleneck region ac. phonon

relaxation from continuum via LO-phonon emission thermalized exciton population in reservoir UP LP E (eV) k(cm-1)

C

∆

Figure 1.10: Scheme of the pumping and scattering processes for cavity polaritons. Image is taken from [38]

ˆ H = α 2L3â † 0â†0â0â0+ ∑ k ~2_k2 2m aˆ † kâk+ α 2L3 ∑ k̸=0 ( 4â†₀ˆa†_kâ0âk+ ˆa†kaˆ−k† â0aˆ0+ â0†â†0âkˆa−k ) . (1.42) Assuming that the total number of particles N is conserved and large, we can write the normalization relation N = â†0â0+

∑ k̸=0ˆa†kâk. Finally, we get: ˆ H = 1 2αnN + ∑ k ~2_k2 2m aˆ † kâk+ 1 2αn ∑ k̸=0 ( 2ˆa†_kâk+ ˆa_k†ˆa†_−k+ âkˆa−k ) . (1.43)

This Hamiltonian could be diagonalized by Bogoliubov linear transformations[39]: ˆ ak= ukˆbk+ v−kˆb†_−k, ˆ a†_k= ukˆb†_k+ v−kˆb−k. (1.44) Two parameters uk, v−k can be written as:

uk, v−k =± ( ~2_k2/_{2m + αn} 2ε(k) ± 1 2 )1/2 , (1.45)

where the dispersion of excitation spectrum is:

ε(k) =± [ αn m ~ 2_k2₊ ( ~2_k2 2m )2]1/2 . (1.46)

As one can see, there are two dispersion branches - with positive and negative values of energy. The dispersion is plotted on g. 1.11.

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-1,0 -0,5 0,0 0,5 1,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 E ( m e V ) k (m -1 ) Bogoliubov dispersion Non-interacting bosons

Figure 1.11: Black line shows positive and negative branches of Bogoliubov disper-sion, when red line is the dispersion for non-interacting particles.

In new basis the Hamiltonian1.43has a simple form: ˆ

H = E0+

∑

k̸=0

ε(k)ˆb†_kˆbk, (1.47)

which is a Hamiltonian for a system of non-interacting quasi-particles, so-called bogolons, which could be considered as collective excitations. By this transforma-tion, the ground state of bosons at zero temperature turned to a vacuum state for bogolons. The energy of the ground state E0in the lowest order calculations is equal

to 1

2αnN. Bogoliubov excitation spectrum rstly was observed in 1999 in the work

[40] on atomic BEC.

The dispersion law 1.46 for bogolons with small wave-vectors is linear by k (sound-like, g.1.11). By this we can derive the eective sound velocity in the condensate:

c =

√

αn

m. (1.48)

In opposite case of large wave-vectors the dispersion becomes parabolic and has a free-particle form.

Interaction constant α for excitons and cavity polaritons is coming mainly from exciton-exciton exchange interaction. It is positive, which means repulsive interac-tion between particles, and can be written in a form [41]:

α = 6EbX2Ca2B. (1.49)

Here Eb is an exciton binding energy, XC is the excitonic fraction of polariton

and aB is a Bohr radius.

As it was said, Bogoliubov theory considered an innite homogeneous Bose gas. In 1961 Gross[42] and Pitaevskii[43] independently have extended the Bogoliubov

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1.3. Bose-Einstein condensates 21 theory to the case of nonuniform condensate. The main idea was to describe by a classical eld not only the ground state, but all other states in the system:

ψ(r, t) = ψ0(r)e−iµt+

∑

k̸=0

ψke−i[k·r−(µ±ω)t]. (1.50)

Putting 1.50 to the Heisenberg equation of motion one will get:

i~d dtψ(r, t) = [ − ~ 2m∇ 2_{+ α}_{|ψ(r, t)|}2 ] ψ(r, t). (1.51)

This equation is called Gross-Pitaevskii equation and it is widely used to describe inhomogeneous coherent Bose gases. However, 1.53does not take into account the decoherence and spontaneous scattering between states.

Finally, since we consider quasiparticles with nite lifetime τ, we can intro-duce phenomenologically the corresponding term to1.53. Also, often the pumping term P is introduced into the equation which simulates the coherent quasi-resonant pumping. The nal Gross-Pitaevskii equation could be written as:

i~d dtψ(r, t) = [ − ~ 2m∇ 2_{+ α}_{|ψ(r, t)|}2₋ i~ 2τ ] ψ(r, t) + P. (1.52)

Polaritons are formed by excitons and photons which total angular momentum projection on Z could take two values: ±1. The polariton-polariton interaction is coming from their excitonic fractions and it was found to be spin-anisotropic[44,45], notably co-polarized excitons interact strongly, and their interaction is repulsive, while polaritons with opposite spins are slightly mutually attracted. Because of that, it is reasonable to consider a two-component spinor polariton condensate. Let's designate σ± _{polarized states as the ψ}_± _{functions. We will discuss the nature}

of exciton-exciton interaction later, however, we will introduce two interaction con-stants: α1 for a triplet state and α2for a singlet state. Taking into account TE-TM

splitting (eq. 1.28) we can rewrite the Gross-Pitaevskii equation in a spinor form:

i~∂ψ± ∂t =− ~2 2m∗∆ψ±+ α1|ψ±| 2 ψ_±+ α2|ψ∓|2ψ±+ βLT(∂y∓i∂x)2ψ∓− i~ 2τψ±+ P±. (1.53) This is the most general form of the Gross-Pitaevskii equation that will be used throughout this thesis.

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Chapter 2

Coherent interactions between

phonons and exciton or

exciton-polariton condensates

Contents

2.1 Basics of exciton-phonon interaction . . . 29

2.1.1 Phonons . . . 29

2.1.2 Exciton-phonon scattering . . . 30

2.1.3 Phonoritons . . . 32

2.2 Exciton-phonon interaction in 2D. . . 33

2.2.1 SAW . . . 34

2.2.2 Acoustic cavities and waveguides . . . 37

2.3 Coherent interactions between phonons and exciton or exciton-polariton condensates . . . 38

2.3.1 Formalism. . . 39

2.3.2 Analytical solution . . . 40

2.3.3 Wavevector dependence . . . 44

2.3.4 Conclusions . . . 47

The interactions between dierent quasiparticles in solids often lead to interest-ing eects and sometimes, in the case of strong interaction, could provide quasipar-ticles of a new type. For example, exciton-polaritons or cavity polaritons appear in the microcavities [1] as the elementary excitations formed by the coupling of excitons and photons. However, in the solids there is another type of collective excitations which is related to collective oscillations of the crystal lattice and which is usually described in terms of quasiparticles called phonons. Phonons can interact both with photons and with excitons.

The interaction between lattice oscillations and light has been studied for a long time, and it would be fare to say, that history of its investigation has its beginning in the works of Rayleigh[2, 3]. In spite of the fact, that in these works Rayleigh considered the elastic scattering of light by disordered media (gas) and not by a crystal, it was the rst successful attempt to describe the scattering of electromagnetic waves by atoms.

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Later, in the beginning of XX century the inelastic scattering was described and observed in two independent works. First - done by Raman - was performed in liquids[4]. The author has shown the appearance of additional spectral lines in the scattered light, which come from the vibrational and rotational excitations of molecules. At the same time, Landsberg and Mandelstam worked on the inelastic scattering by solids. Predicted theoretically by Mandelstam and Brillouin[5], this eect was nally observed and described in the work[6]. However, the eect oc-curred to be much stronger than it was expected. It has indicated the fundamental dierence between solids and liquids, and revealed the necessity of individual theory of oscillations in solids.

If the frequency of the electro-magnetic wave, propagating in the crystal, is close to the exciton frequency, it is necessary to speak about exciton-polariton and consider the excitonic part as well. Carriers, composing the exciton, could eectively interact with lattice oscillations. Moreover, polaritons demonstrate ecient energy relaxation because of their excitonic part.

Because the exciton-phonon interaction is weak relatively to exciton-photon in-teraction, in general, it causes only Brillouin scattering. However, it was predicted by A. L. Ivanov et al.[7], that in a case of strong electro-magnetic wave, some sort of strong coupling between exciton-polaritons and phonons could occur, giving rise to new quasi-particle - phonoriton. Several experimental works have shown indi-rect evidence of this eect[8,9], but because in the bulk both exciton-photon and exciton-phonon interactions are not so strong, directly the eect still was not ob-served.

The situation gets better with the reducing of the dimensionality. It was recently shown[10] that 2D cavity polaritons could demonstrate a signicant reconstruction of their dispersion when interacting with a strong 2D surface acoustic wave. The theory of interaction between the 2D polaritons and a 2D acoustic wave was developed by A. L. Ivanov[11]. The phonon eld has been treated as an external classical eld, and its reduced dimensionality was obtained due to strong coherent external 2D pumping. However, it could be reduced also by embedding an acoustic cavity or an acoustic waveguide inside the optical cavity, like in [12]

In our work[13] we consider the interaction between a condensate of quantum well excitons or cavity exciton-polaritons and a coherent phonon eld, possessing the same dimensionality. We develop a theory of interaction between phonons and bogolons (elementary excitations of the condensate) and we show that at some condi-tions the strong coupling regime could be obtained, resulting in a strong modication of the dispersion, and even in the appearance of a "roton instability" region.

This chapter will be organized as follows. In the rst part I will describe the ba-sics of phonons and of electron (exciton)-phonon interaction. Further, I will discuss the recent theoretical and experimental works performed on the strong acousto-polariton interaction in the systems with reduced dimensionality. And nally, I will explain the main idea of our work, introduce our theoretical model and describe the results that we have obtained.

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2.1. Basics of exciton-phonon interaction 29

2.1 Basics of exciton-phonon interaction

2.1.1 Phonons

Let us consider an innite 1D lattice of equal atoms mutually bound by elastic force described by a coecient C. One can write the following equation to describe the motion of n-th atom:

md

2_u

n

dt2 =−2Cun+ C(un+1+ un−1), (2.1)

where unis the n-th atoms coordinate. This is a wave equation, so we can make

a transformation from real coordinates un, which describe the position of each atom,

to normal coordinates Uq, which describe dierent waves with dierent wavevectors:

un=

∑

q

Uqeiqnd. (2.2)

Then, the wave equation will read:

md

2_U

q

d t2 = 2C(cos qd− 1)Uq. (2.3)

The solutions of this equation can be easily found:

Uq= Aqeiωqt, ωq =

√ 2C

m(1− cos qd). (2.4)

So, the normal coordinates describe dierent vibrational modes, whose frequency obeys its own dispersion law. By this approach, one can treat these oscillations as a set of quasiparticles. This concept was rst introduced by Igor Tamm in 1930 and the quasiparticles were called phonons by Yakov Frenkel.

Now, if we consider that 1D lattice is composed by 2 dierent types of atoms with masses m1,2 with the spacing between two equal atoms a, we will obtain two

dierent dispersion branches:

ω_±2 = C ( 1 m1 + 1 m2 ) ± C √( 1 m1 + 1 m2 )2 − 4 sin2(qa/2) m1m2 . (2.5)

For small values of wave-vector q we can write:

ω₋= √ C 2(m1+ m2) qa, ω+= √ 2C(m1+ m2) m1m2 . (2.6)

The rst dispersion branch is linear by q and the quasiparticles corresponding to this branch are called acoustic phonons. The second curve is nonzero for q = 0 and is called the dispersion branch of optical phonons. The schematic image of the two dispersion branches is shown at2.1. Physically it means, that in the solids with an elementary cell containing more than one atom, a new type of phonons appears.

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Figure 2.1: Optical and acoustic phonon dispersion branches. Image is taken from [14]

Moreover, these phonons could carry signicant amounts of energy even with small wave-vectors.

Mandelstam and Landsberg in their experiment expected to observe the scat-tering of light by acoustic phonons, which should result in additional spectral lines close to the general line. However, they surprisingly obtained satellite lines situated quite far from the main one, which has been interpreted as an experimental evidence of the interaction with the optical phonons.

2.1.2 Exciton-phonon scattering

The common life cycle of an exciton or an exciton-polariton starts with its optical or electrical generation (electrons and holes are formed, and then they bind into excitons) at high-energy states, followed by its energy relaxation and decay. If the energy of hot excitons is high enough, the initial scattering process is mediated mainly by optical phonons. In each scattering act, quite large amounts of energy are exchanged - up to tens meV . The scattering is driven by so called Frohlich inter-action, by the name of Herbert Frohlich - German-Brittish physicist, who described theoretically[15] the interaction between electrons and optical phonons in solids.

In the approximation of long wavelength optical phonons, one can consider the constant lattice oscillation frequency ωLO. By this the energy of interaction between

an electron and LO phonon with wave-vector q [15]:

E_eq_−LO ∼ 4πe

2

a3_k√_2N sin(ωLOt +kr), (2.7)

where the sum goes over all possible phonon states, e is a charge of an electron,

a is a size of an unit cell and N is a number of unit cells in a crystal. The matrix

Collective dynamics of excitons and exciton-polaritons in nanoscale heterostructures

HAL Id: tel-00914332

https://tel.archives-ouvertes.fr/tel-00914332

Dmitri Visnevski

To cite this version:

UNIVERSITE BLAISE PASCAL

-CLERMONT-FERRAND

ECOLE DOCTORALE DES SCIENCES

FONDAMANTALES

THESE DE DOCTORAT

Docteur d'université en sciences

de l'université Blaise Pascal - Clermont-Ferrand

Specialité : Science des matériaux

Dmitri VISNEVSKI

Collective dynamics of excitons

and exciton-polaritons

in nanoscale heterostructures

Acknowledgments

Contents

Chapter 1

Introduction

1.1 Excitons and Exciton-polaritons

1.2 Nanostructures

1.3 Bose-Einstein condensates

(a)

(b)

Ω

X

C

∆

Bibliography

Chapter 2

Coherent interactions between

phonons and exciton or

exciton-polariton condensates

2.1 Basics of exciton-phonon interaction